FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES

Transcription

1 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES WILL BONEY AND RAMI GROSSBERG Abstract. We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1. Let M 0 N be models from K and A be a set. We say that the Galois-type of A over N does not fork over M 0, written A N, iff for all small a A and all small N N, we have that Galois-type of a over N is realized in M 0. Assuming property (E) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a big cardinal. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [BGKV], it is established that, if this notion is an independence notion, then it is the only one. Contents 1. Introduction 2 2. Preliminaries 6 3. Axioms of an independence relation and the definition of forking 9 4. Connecting Existence, Symmetry and Uniqueness The main theorem Getting Local Character The U-Rank Large cardinals revisited 36 References 42 Date: January 3, 2017 Part of this material is based upon work done while the first author was supported by the National Science Foundation under Grant No. DMS AMS 2010 Subject Classification: Primary: 03C48, 03C45 and 03C52. Secondary: 03C55, 03C75, 03C85 and 03E55. 1

2 2 WILL BONEY AND RAMI GROSSBERG 1. Introduction Much of modern model theory has focused on Shelah s forking. In the last twenty years, significant progress has been made towards understanding of unstable theories, especially simple theories (Kim [Kim98] and Kim and Pillay [KiPi97]), N IP theories (surveys by Adler [Ad09] and Simon [Si]), and, most recently, NT P 2 (Ben- Yaacov and Chernikov [BYCh14] and Chernikov, Kaplan, and Shelah [CKS1007]). In the work on classification theory for Abstract Elementary Classes (AECs), such a nicely behaved notion is not known to exist. However, much work has been done towards this goal. Around 2005, homogeneous model theory working under the assumption that there exists a monster model which is sequential homogeneous (but not necessarily saturated as in the first-order sense) and types consists of sets of first-order formulas reached a stage of development that parallels that of first-order model theory in the seventies. There is a Morley-like categoricity theorem (Keisler [Kei71] and Lessmann [Les00]), forking exists (Buechler and Lessman [BuLe03]), and even a main gap is true (Grossberg and Lessmann [GrLe05]). Hyttinen and Kesälä studied a further extension of homogeneous model theory called finitary AECs in [HyKe06] and in [HyKe11]. They established both Morleys categoricity theorem and that non-splitting is a variant of forking under the assumptions of ℵ 0 -stability and what they call simplicity (like our extension property) in a countable language. However, as AECs are much more general the situation for AECs is more complicated. There are classes axiomatized by L ω1,ω that do not fit into the framework of homogeneous model theory: (1) Marcus [Mar75] constructed an L ω1,ω sentence that is categorical in all cardinals but does not have even an ℵ 1 -homogenous model. (2) Hart and Shelah [HaSh323] constructed, for each k < ω, an L ω1,ω sentence ψ k which is categorical in all ℵ n for n k but not categorical in higher cardinals. By the categoricity theorem for finitary AECs [HyKe11], this means that Mod(ψ k ) is not homogeneous as it is not even finitary. In [Sh:h, Chapter N], Shelah explains the importance of classification theory for AECs. At the referee s suggestion, we summarize the argument here, although the truly interested reader should consult the source. As mentioned above, classification theory has become the main focus of model theory. Shelah and other early workers were motivated by purely abstract problems, such as the main gap in [Sh:c]. The machinery used to solve these problems turned out to be very powerful and, about 20 years later, Chatzidikis, Hrushovski, Scanlon, and others discovered deep applications to geometry, algebra, and other fields. However, this powerful machinery was restricted because it only applied to firstorder model theory. This is natural from a logical point of view as first-order logic has many unique features (e. g., compactness), but there are many mathematical

3 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 3 classes that are not first order axiomatizable: we list some in this introduction and in Section 5 and each of Grossberg [Gro02], Baldwin [Bal09], and [Sh:h, Chapter N] contain their own lists. The logic needed to axiomatize each context varies, from L ω1,ω(q) for quasiminimal classes to L R +,ω for torsion R-modules. Varying the substructure relation (e. g., subgroup vs. pure subgroup) complicates the picture further. A unifying perspective is given by AECs and Shelah began their classification (and their study) in the late 1970s. Here again questions of number of nonisomorphic models have formed the basic test questions. The most central one here is Shelah s Categoricity Conjecture; Shelah proposed this conjecture for L ω1,ω in the late seventies as a way to measure the development of the relevant classification theory. At present, there are many partial results that approximates this conjecture and harder questions for AECs. Despite an estimated of more than 2,000 published pages, the full conjecture is not within reach of current methods, in contrast to the existence of relatively simple proofs of the conjecture for the cases of homogeneous models and finitary AECs. Due to the lack of compactness and syntax, extra set-theoretic assumptions (in addition to new techniques) have been needed to get these results; the strong Devlin-Shelah diamonds on successors in [Sh576] and large cardinals in [MaSh285] are excellent examples of this. Differing from Shelah, our vision is that model-theoretic assumptions (especially tameness and type-shortness here) will take the place of set-theoretic ones. The hope here is two-fold: first, that, although these assumptions don t hold everywhere, they can be shown to hold in many natural and, second, that these model-theoretic assumptions are enough to develop a robust classification theory. This paper (and follow-ups by Boney, Grossberg, Kolesnikov, and Vasey) provide evidence for the first hope and examples described in Section 5 provide evidence for the first. In [Sh394], Shelah introduced analogues of splitting and strong splitting for AECs. Building on this, Shelah [Sh576], and Grossberg, VanDieren, and Villaveces [GVV], Grossberg and VanDieren [GV06c, GV06a] used tameness to prove an upward categoricity transfer. This helped cement tameness as an important property in the classification of AECs. Working in a stronger context, Makkai and Shelah [MaSh285] studied the case when a class is axiomatized by an L κ,ω theory and κ is strongly compact. They managed to obtain an eventual categoricity theorem by introducing a forking-like relation on types. In this particular case, Galois types (defined in 2) can be identified with complete set of formulas taken from a fragment of L κ,κ. In their paper, Makkai and Shelah assumed not only that κ is strongly compact but also that the class of structures is categorical in some λ + where λ ℶ (2 κ ) +. Our paper is an extension and generalization of the above results of Makkai and Shelah, but with assumptions closer to those of Grossberg and VanDieren. We introduce a notion that, like the one from [MaSh285], is an analogue of the

4 4 WILL BONEY AND RAMI GROSSBERG first order notion of coheir. One of our main results is that, given certain model theoretic assumptions, this notion is in fact an independence notion. Theorem (5.1). Let K be an AEC with amalgamation, joint embedding, and no maximal models. If there is some κ > LS(K) so that (1) K is fully < κ-tame; (2) K is fully < κ-type short; (3) K doesn t have an order property; and (4) satisfies existence and extension, then is an independence relation. Sections 2, 3, and 4 give precise definitions and discussions of the terms in the theorem. After proving the main theorem in Section 5, we delve into a discussion of the assumptions used; the difficulties of the proof compared with first-order; and some examples of where the theorem can be applied. At the referee s suggestion, we have included a list of the other main results in the paper; these all take place under the hypothesis of amalgamation, joint embedding, and no maximal models and κ > LS(K). Theorem. If K has no weak κ-order property, is categorical in λ κ, and Galois stable in κ, then κ ω( ) = ω; see Theorem 6.8. If K is fully < κ-tame and -type short, doesn t have the weak κ-order property, is categorical in λ > κ, and satisfies existence and extension, then K [κ,λ) has unique limit models in each cardinality; see Corollary If is an independence relation, then the forking rank characterizes nonforking on ordinal-ranked Galois types; see Theorem 7.7. If κ is strongly compact and K is categorical in λ = λ <κ, then is an independence relation with superstable-like local character; see Theorem 8.2.(3). The main theorem also generalizes the work of [Sh472]. This has several advantages over previous work. (1) We generalize Makkai and Shelah [MaSh285] in two key ways. First, we replace L κ,ω classes and syntactic types by AECs and Galois types. This allows the results to apply to classes not axiomatizable in L κ,ω, such as quasiminimal excellent classes (which require the Q quantifier) and the example of Baldwin, Eklof, and Trlifaj [BET07]. Second, we replace large cardinal axioms by purely model theoretic hypotheses: tameness and type shortness. Section 5 gives some ZFC examples of AECs with these properties. Together, tameness and type shortness give a locality condition for when an injection with domain not necessarily a model can be extended into a K embedding; see [Bon14b]. 3 for a longer discussion. (2) When reduced to the special case where K has a first-order axiomatization by a complete theory, the independence relation we introduce is coheir,

5 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 5 which is equivalent to first-order forking in the stable case we consider. This is unlike the relatives of splitting and strong splitting used by [HyKe06] and [GrLe02]. This allows us to mimic some first order arguments; see Section 6. (3) Motivated by a test question of Grossberg 1, Shelah [Sh576], [Sh600], and [Sh705] and Jarden and Shelah [JrSh875] have dealt with the problem whether I(λ, K) = 1 I(λ +, K) < 2 λ+ implies existence of model of cardinality λ ++. While this question is still open (even under strong set-theoretic assumptions), Shelah managed to get several approximations. For this, he needed to discover and develop a very rich conceptual infrastructure that occupies more than 500 pages. One of the more important notions is that of good λ-frame. This is a forking-like relation defined using Galois-types over models of cardinality λ. Our approach is orthogonal to Shelah s recent work on good λ-frames and we manage to obtain a forking notion on the class of all models above a natural threshold size, instead of models of a single cardinality. Instead of using I(λ +n, K) < 2 λ+n for all 1 n < ω, we assume the lack of an order property, which follows from few models in a single big cardinal. Unlike Shelah, our treatment does not make use of diamond-like principles as we work in ZFC. More comparison is given after Definition 3.3. The main theorem allows us to shed light on other questions in the classification theory of AECs, especially concerning superstability. Superstability in AECs suffers from schizophrenia [Sh:h, p. 19]. The main result shows a connection between two conditions, namely that a local character κ ( ) = ω implies uniqueness of limit models; see Section 6. Another question involves properties of ranks (see [Vi06, Question 3]) and Section 7 introduces a rank based on the properties of the independence relation that is ordinal valued exactly when a local character for holds. Unfortunately, there is no free lunch and we pay for this luxury. Our payment is essentially in assuming tameness and type-shortness. As was shown by Boney in [Bon14b], these assumptions are corollaries of certain large cardinal axioms, including the one assumed by Makkai and Shelah; indeed, more recent work of Boney and Unger [BoUn] show that assuming every AEC is tame is equivalent to a large cardinal axiom. However, many natural AECs are tame, and it seems to be plausible that tameness and type shortness will be derived in the future from categoricity above a certain Hanf number that depends only on LS(K). After circulating early drafts of this paper, some of our results were used by Sebastien Vasey [Vasb] and [Vasc], e. g., Theorem 6.8 is crucial in his proof of superstability from categoricity. Also Theorem 8.2.(2) is used in a forthcoming article by Grossberg and Vasey [GVas]. 1 This first appeared in Grossberg s 1981 MSc thesis.

6 6 WILL BONEY AND RAMI GROSSBERG Section 2 gives the necessary background information for AECs. Section 3 gives a list of common axioms for independence relations and defines the forking relation that we will consider in this paper. Section 4 gives a fine analysis of when parameterized versions of the axioms from Section 3 hold about our forking relation. Section 5 gives the global assumptions that make our forking relation an independence relation. Section 6 introduces a notion that generalizes heir and deduces local character of our forking from this and categoricity. Section 7 introduces a U rank and shows that it is well behaved. Section 8 continues the study of large cardinals from [Bon14b] and shows that large cardinal assumptions simplify many of the previous sections. This paper was written while the first author was working on a Ph.D. under the direction of Rami Grossberg at Carnegie Mellon University, and he would like to thank Professor Grossberg for his guidance and assistance in his research in general and relating to this work specifically. He would also like to thank his wife Emily Boney for her support. A preliminary version of this paper was presented in a seminar at Carnegie Mellon and we appreciate comments from the participants, in particular Jose Iovino. We would also like to thank John Baldwin, Adi Jarden, Sebastien Vasey and Andres Villaveces. We are very grateful to the referee for comments on several versions of this paper, the referee s reports significantly improved our paper. 2. Preliminaries The definition of an Abstract Elementary Class was first given by Shelah in [Sh88]. The definitions and concepts in this section are all part of the literature; in particular, see the books by Baldwin [Bal09] and Shelah [Sh:h], the article by Grossberg [Gro02], or the forthcoming book by Grossberg [Gro1X] for general information. Definition 2.1. We say that (K, K ) is an Abstract Elementary Class iff (1) There is some language L = L(K) so that every element of K is an L- structure; (2) K is a partial order on K; (3) for every M, N K, if M K N, then M is an L-substructure of N; (4) (K, K ) respects L isomorphisms: if f : N N is an L isomorphism and N K, then N K and if we also have M K with M K N, then f(m) K N ; (5) (Coherence) if M 0, M 1, M 2 K with M 0 K M 2, M 1 K M 2, and M 0 M 1, then M 0 M 1 ; (6) (Tarski-Vaught axioms) suppose M i K : i < α is a K -increasing continuous chain, then (a) i<α M i K and, for all i < α, we have M i K i<α M i ; and

7 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 7 (b) if there is some N K so that, for all i < α, we have M i K N, then we also have i<α M i K N. (7) (Lowenheim-Skolem number) LS(K) is the first infinite cardinal λ L(K) such that for any M K and A M, there is some N K M such that A N and N A + λ. Remark 2.2. As is typical, we drop the subscript on K when it is clear from context and abuse notation by calling K an AEC when we mean (K, K ) is an AEC. Also, we follow the convention of Shelah that, for M K, we denote the cardinality of its universe by M. Also, in this paper, K is always an AEC that has no models of size smaller than the Lowenheim-Skolem number. We will briefly summarize some of the necessary basic notations, definitions, and results for AECs; the references contain a more detailed description and development. Definition 2.3. (1) A K embedding from M to N is an injective L(K)-morphism f : M N so f(m) K N. (2) K λ = {M K : M = λ} K λ = {M K : M λ} (3) K has the amalgamation property (AP) iff for any M N 0, N 1 K, there is some N K and f i : M N i such that N 1 f 1 N M commutes. (4) K has the joint embedding property (JEP) iff for any M 0, M 1 K, there is some M K and f i : M i M. (5) K has no maximal models iff for all M K, there is N K so M N. Note that it is a simple exercise to show that, if K has the joint embedding property, then having arbitrarily large models is equivalent to having no maximal models. We will use the above three assumptions in tandem throughout this paper. This allows us to make use of a monster model, as in the complete, first order setting; [Gro1X, Section 4.4] gives details. The monster model C is of large size and is universal and model homogeneous for all models that we consider. As is typical, we assume all elements come from the monster model. We use a monster model to streamline our treatment. However, amalgamation is the only one of the properties that is crucial because it simplifies Galois types. N 0 f 2

8 8 WILL BONEY AND RAMI GROSSBERG Joint embedding and no maximal models are rarely used; one major exception is Proposition 5.3 in the discussion of the order property. After giving the definition of nonforking in the next section, we briefly detail the differences when we are not working in the context of a monster model. In AECs, a consistent set of formulas is not a strong enough definition of type; any of the examples of non-tameness will be an example of this and it is made explicit in [BK09]. However, Shelah isolated a semantic notion of type in [Sh300] that Grossberg suggested to call Galois type in [Gro02]. In his book [Sh:h], Shelah calls this orbital type. This notion replaces the first order notion of type for AECs. Crucially, we allow Galois types of infinite lengths. Definition 2.4. Let K be an AEC, λ LS(K), and I be a nonempty set. (1) Let M in K and a i C : i I be a sequence of elements. The Galois type of a i C : i I over M is denoted gtp( a i C : i I /M) and is the orbit of a i : i I under the action of automorphisms of C fixing M. That is, a i C : i I and b i : i I have the same Galois type over M iff there is f Aut M C so that f(a i ) = b i for all i I. (2) For M K, gs I (M) = {gtp( a i : i I /M) : a i C for all i I}. (3) Suppose p = gtp( a i : i I /M) gs I (M) and N M and I I. Then, p N gs I (N) is gtp( a i : i I /N) and p I gs I (M) is gtp( a i : i I /M). (4) Given a Galois type p gs I (M), then the domain of p is M and the length of p is I. (5) If p = gtp(a/m) is a Galois type and f Aut C, then f(p) = gtp(f(a)/f(m)). Remark 2.5. We sometimes write that the type of two sets (say X and Y ) are equal; given the above definitions, this really means there is some enumeration X = x i : i I and Y = y i : i I so that the types of the sequences are equal. If we reference some x X and the corresponding part of Y, then this refers to y Y indexed by the same set that indexes x; that is, y = y i : i I and x i x. Along with types comes a notion of saturation, called Galois saturation. A degree of Galois saturation will be necessary when we deal with our independence relation, so we offer a definition here. Additionally, we include a lemma of Shelah that characterizes saturation by model homogeneity. Definition 2.6. (1) A model M K is µ-galois saturated iff for all N M such that N < µ and p gs(n), we have that p is realized in M. (2) A model M K is µ-model homogeneous iff for all N M and N N such that N < µ, there is f : N N M. Lemma 2.7 ( [Sh576] ). Let λ > LS(K) and M K and suppose that K has the amalgamation property. Then M is λ-galois saturated iff M is λ-model homogeneous.

9 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 9 We conclude the preliminaries by recalling two locality properties that are key for this paper: tameness and type shortness. Tameness was first isolated by Grossberg and VanDieren [GV06b, Definition 3.2], although a weaker version had been used by Shelah [Sh394] in the midst of a proof. Later, Grossberg and VanDieren [GV06c] [GV06a] showed that a strong form of Shelah s Categoricity Conjecture holds for tame AECs. Type shortness was defined by the first author in [Bon14b, Definition 3.3] as a dual property for tameness. There, type shortness and tameness were derived from large cardinal hypotheses. Recall that P κ I = {I 0 I : I 0 < κ} and P κm = {M M : M < κ}. Definition 2.8. (1) K is (< κ, λ)-tame for θ-length types iff, for all p, q gs I (M) with M = λ and I = θ, we have p = q iff, for all M PκM, p M = q M. (2) K is (< κ, λ)-type short for θ-sized domains iff, for all p, q gs I (M) with I = λ and M = θ, we have p = q iff, for all I 0 P κ I, p I 0 = q I 0. (3) K is fully < κ-tame and -type short iff it is (< κ, λ)-tame for θ-length types and (< κ, λ)-type short for θ-sized domains for all θ and all λ κ. We parameterize the properties to get exact results in Proposition 4.1, but the reader can focus on the fully < κ-tame and -type short case if desired. Note, by [Bon14b, Theorem 3.5], the type shortness implies the tameness. However, we include both hypotheses for clarity. 3. Axioms of an independence relation and the definition of forking The following hypothesis and definition of non-forking is central to this paper: Hypothesis 3.1. Assume that K has no maximal models and satisfies the λ-joint embedding and λ-amalgamation properties for all λ LS(K). Fix a cardinal κ > LS(K). The nonforking is defined in terms of this κ and all subsequent uses of κ will refer to this fixed cardinal, until Section 8. If we refer to a model, tuple, or Galois type as small, then we mean its size is < κ, its length is of size < κ, or both its domain and its length are small. Definition 3.2. Let M 0 N be models and A be a set. We say that gtp(a/n) does not fork over M 0, written A N, iff for all small a A and all small N N, we have that tp(a/n ) is realized in M 0. We call this κ-coheir or < κ-satisfiability. That is, a type does not fork over a base model iff all small approximations to it, both in length and domain, are realized in the base model. This definition is a relative of the finite satisfiability also known as coheir notion of forking that is extensively studied in stable theories. It is an AEC version of the non-forking defined in Makkai and Shelah [MaSh285, Definition 4.5] for categorical L κ,ω theories when κ is strongly compact.

10 10 WILL BONEY AND RAMI GROSSBERG We now list the properties that, our nonforking notion will have. These properties can be thought of as axiomatizing an independence relation. The ones listed below are commonly considered and are similar to the properties that characterize nonforking in first order, stable theories, although this list is most inspired by [MaSh285, Proposition 4.4]. However, many of these properties have been changed because we require the bottom and right inputs to be models. This is similar to good λ-frames, which appear in [Sh:h, Section II.2], although we don t require the parameter set A to be a singleton and we allow the sets and models to be of any size. The properties we introduce are heavily parameterized. The interesting and hard to prove properties Existence, Extension, Uniqueness, and Symmetry are each given with parameters among λ, µ, χ, and θ. These parameters allows us to conduct a fine analysis of exactly what assumptions are required to derive these properties. The order of these parameters is designed to be as uniformized as possible: if appropriate, when referring to A N, the λ refers to the size of A, µ refers to the size of M 0, and χ refers to the size of N. If we write a property without parameters, then we mean that property for all possible parameters. Definition 3.3. Fix an AEC K. Let be a ternary relation on models and sets so that A N implies that A is a subset of the monster model and M 0 N are both models. We say that is an independence relation iff it satisfies all of the following properties for all cardinals referring to sets and all cardinals that are at least κ when the cardinal refers to a model. (I) Invariance Let f Aut C be an isomorphism. Then A N implies f(a) (M) Monotonicity If A N and A A and M 0 M 0 N N, then A N. (T ) Transitivity If A M 0 (C) <κ Continuity N and M 0 N with M 0 M 0, then A N. M 0 M 0 f(m 0 ) f(n). (a) If for all small A A and small N N, there are M 0 M 0 and N N N such that M 0 N and A N, then A N. (b) If I is a κ-directed partial order and A i, M i 0 i I are increasing such that A i N for all i I, then i I A i M i 0 M 0 i I i N.

11 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 11 (E) (λ,µ,χ,θ) (a) Existence Let A be a set and M 0 be a model of sizes λ and µ, respectively. Then A M 0. (b) Extension Let A be a set and M 0 and N be models of sizes λ, µ, and χ, respectively, so that M 0 N and A N. If N + N of size θ, then there is A so A N + and gtp(a /N) = gtp(a/n). M 0 (S) (λ,µ,χ) Symmetry Let A 1 be a set, M 0 be a model, and A 2 be a set of sizes λ, µ, and χ, respectively, so that there is a model M 2 with M 0 M 2 and A 2 M 2 such that A 1 M 2. Then there is a model M 1 M 0 that contains A 1 such that M 0 A 2 M 1. M 0 (U) (λ,µ,χ) Uniqueness Let A and A be sets and M 0 N be models of sizes λ, λ, µ, and χ, respectively. If gtp(a/m 0 ) = gtp(a /M 0 ) and A N and A tp(a/n) = tp(a /N). M 0 N, then A discussion of these axioms and their relation to other nonforking notions is in order. As mentioned in the introduction, this notion is somewhat orthogonal to Shelah s notion of good λ-frames (see [Sh:h, Definition II.2.1]). While both attempt to axiomatize a nonforking relation, we allow greater generality by considering Galois types of arbitrary length over all models of a sufficiently large size. In contrast, Shelah deals with a subclass of unary Galois types only over a fixed size λ. On the other hand, since good λ-frames attempt to axiomatize superstability (rather than stability as we do here), good λ-frames have stronger continuity and local character properties. Notice that the Existence property implies that M 0 K κ is κ-galois saturated when is κ-coheir. However, this is not a serious restriction in comparison with other work. In other cases where AECs have a strong nonforking notions, some level of categoricity is assumed; see Makkai and Shelah [MaSh285], Shelah [Sh:h, Section II.3], and Vasey [Vasa]. The categoricity hypotheses are used to ensure large amounts of Galois saturation, as we have here. Indeed, Theorem 5.4 below shows that categoricity in some λ = λ <κ implies that all sufficiently large models are κ-galois saturated. The monotonicity and invariance properties are actually necessary to justify our formulation of nonforking as based on Galois types. Without them, whether or

12 12 WILL BONEY AND RAMI GROSSBERG not a Galois type doesn t fork over a base model could depend on the specific realization of the chosen type. Since these properties are clearly satisfied by our definition of nonforking, this is not an issue. The axiom (E) (λ,µ,χ,θ) combines two notions. The first is Existence: that a Galois type does not fork over its domain. This is similar to the consequence of simplicity in first order theories that a type does not fork over the algebraic closure of its domain. As mentioned above, in this context, existence is equivalent to every model being κ-galois saturated. In the first order case, where finite satisfiability is the proper analogue of our non-forking, existence is an easy consequence of the elementary substructure relation. In [MaSh285], this holds for < κ satisfiability, their nonforking, because types are formulas from L κ,κ and, due to categoricity, the strong substructure relation is equivalent to Lκ,κ. The second notion is the extension of nonforking Galois types. In first order theories (and in [MaSh285]), this follows from compactness, but is more difficult in a general AEC. We have separated these notions for clarity and consistency with other sources, but could combine them in the following statement of (E) (λ,µ,µ,χ) : Let A be a set and M 0 and N be models of sizes λ, µ, and χ, respectively, so that M 0 N. Then there is some A so that gtp(a /M 0 ) = gtp(a/m 0 ) and A N. M 0 As an alternative to assuming (E), and thus assuming all models are κ-galois saturated, we could simply work with the definition and manipulate the nonforking relationships that occur. This is the strategy in Section 6. In such a situation, κ-galois saturated models, which will exist in λ <κ, will satisfy the existence axiom. The relative complexity of the symmetry property is necessitated by the fact that the right side object is required to be a model that contains the base. If the left side object already satisfied this, then there is a simpler statement. Proposition 3.4. If (S) (λ,µ,χ) holds, then so does the following (S ) (λ,µ,χ) Let M, M 0, and N be models of size λ, µ, and χ, respectively so that M 0 N and M 0 M. Then M N iff N M. In first order stability theory, many of the key dividing lines depend on the local character κ(t ), which is the smallest cardinal so that any type doesn t fork over some subset of its of domain of size less than κ(t ). The value of this cardinal can be smaller than the size of the theory, e.g. in an uncountable, superstable theory. However, since types and nonforking occur only over models, the smallest value the corresponding cardinal could take would be LS(K) +. This is too coarse for many situations, although see Boney and Vasey [BoVa] for a comparison. Instead, we follow [ShVi635], [Sh:h, Chapter II], and [GV06b] by defining a local character cardinal based on the length of a resolution of the base rather than the size of cardinals. As different requirements appear in different places, we give two definitions

13 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 13 of local character: one with no additional requirement, as in [Sh:h, Chapter II], and one requiring that successor models be universal, as in [ShVi635] and [GV06b]. Definition 3.5. κ α ( ) = min{λ REG { } : for all regular µ λ and all increasing, continuous chains M i : i < µ and all sets A of size less than α, there is some i 0 < µ so A Mi0 i<µ M i } κ α( ) = min{λ REG { } : for all µ = cf µ λ and all increasing, continuous chains M i : i < µ with M i+1 universal over M i which is κ-saturated and all sets A of size less than α, there is some i 0 < µ so A Mi0 i<µ M i } In either case, if we omit α, then we mean α = ω. In Section 6, we return to these properties and find a natural, sufficient condition that implies that κ ( ) = ω. Although we do not do so here, the notions of κ-coheir makes sense in AECs with some level of amalgamation but without the full strength of a monster model. In such an AEC, the definition of the Galois type of A over N must be augmented by a model containing both; that is, some M K so A M and N M. We denote this type gtp(a/n, M). Similarly, we must add this fourth input to the nonforking relation that contains all other parameters. Then A N iff M 0 N M and M 0 A M and all of the small approximations to the Galois type of A over N as computed in M are realised in M 0. The properties are expanded similarly with added monotonicity for changing the ambient model M and the allowance that new models that are found by properties such as existence or symmetry might exist in a larger big model N. All theorems proved in this paper about nonforking only require amalgamation, although some of the results referenced make use of the full power of the monster model. We end this section with an easy exercise in the definition of nonforking that says that A and N must be disjoint outside of M 0. Proposition 3.6. If we have A N, then A N M 0. Proof: Let a A N. Since N is a model, we can find a small N N that contains a. Then, by the definition of nonforking, gtp(a/n ) must be realized in M 0. But since a N, this type is algebraic so the only thing that can realize it is a. Thus, a M Connecting Existence, Symmetry and Uniqueness In this section, we investigate what AEC properties cause the axioms of our independence relation to hold; recall that we are working under Hypothesis 3.1 M

14 14 WILL BONEY AND RAMI GROSSBERG that K is an AEC with amalgamation, joint embedding, and no maximal models and that κ > LS(K) is fixed. The relations are summarized in the proposition below. Proposition 4.1. Suppose that K doesn t have the weak κ-order property and is (< κ, λ + χ)-type short for θ-sized domains and (< κ, θ)-tame for < κ length types. Then, for the κ-coheir relation, (1) (E) (χ,θ,θ,λ) implies (S) (λ,θ,χ). (2) (S) (<κ,θ,<κ) implies (U) (λ,θ,χ). Recall Definition 2.8 for tameness and type shortness. This proposition and the lemma used to prove it below rely on an order property. Definition 4.2. K has the weak κ-order property iff there are lengths α, β < κ, a model M K <κ, and types p q gs α+β (M) such that there are sequences a i α C : i < κ and b i β C : i < κ such that, for all i, j < κ, i j = gtp(a i b j /M) = p i > j = gtp(a i b j /M) = q This order property is a generalization of the first order version to our context of Galois types and infinite sequences. This is one of many order properties proposed for the AEC context and is similar to 1-instability that is studied by Shelah in [Sh1019] in the context of L θ,θ theories where θ is strongly compact. The adjective weak is in comparison to the (< κ, κ)-order property in Shelah [Sh394, Definition 4.3]. The key difference is that [Sh394] requires the existence of ordered sequences of any length (i.e. the existence of a i, b i : i < δ for all ordinals δ), while we only require a sequence of length κ. We discuss the implications of the weak κ-order property in the next section. For now, we use it s negation to prove the following result, similar to one in [MaSh285, Proposition 4.6], based on first order versions due to Poizat and Lascar. Lemma 4.3. Suppose K is an AEC that is (< κ, θ)-tame for < κ length types and doesn t have the weak κ-order property. Let M 0 M, N such that M 0 = θ and let a, b, a <κ C such that b N and a M. If then gtp(ab/m 0 ) = gtp(a b/m 0 ). gtp(a/m 0 ) = gtp(a /M 0 ) and a N and b M Proof: Assume for contradiction that gtp(ab/m 0 ) gtp(a b/m 0 ). We will build sequences that witness the weak κ-order property. By tameness, there is some M M 0 of size < κ such that gtp(ab/m ) gtp(a b/m ). Set p = gtp(ab/m ) and q = gtp(a b/m ). We will construct two sequences a i l(a) M 0 : i < κ and b i l(b) M 0 : i < κ by induction. We will have, for all i < κ (1) a i b p;

15 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 15 (2) a i b j q for all j < i; (3) ab i q; and (4) a i b j p for all j i. Note that, since b i l(b) M 0, (3) is equivalent to a b i q. This is enough: (2) and (4) are the properties necessary to witness the weak κ-order property. Construction: Let i < κ and suppose that we have constructed the sequences for all j < i. Let N + N of size < κ contain b, M, and {b j : j < i}. Because a N, there is some a i M 0 that realizes gtp(a /N + ). This is witnessed by f Aut N +C with f(a) = a i. Claim: (1) and (2) hold. f fixes M and b, so it witnesses that gtp(ab/m ) = gtp(a i b/m ). Similarly, it fixes b j for j < i, so it witnesses q = gtp(ab j /M ) = gtp(a i b j /M ). Claim Similarly, pick M + M of size < κ to contain M, a, and {a j : j i}. Because b M, there is b i M 0 that realizes gtp(b/m + ). As above, (3) and (4) hold. Now we are ready to prove our theorems regarding when the properties of hold. The first four properties always hold from the definition of nonforking. Proposition 4.4. satisfies (I), (M), (T ), and (C) <κ. To get the other properties, we have to rely on some degree of tameness, type shortness, no weak order property, and the property (E). Proof of Proposition 4.1: (1) Suppose (E) (χ,θ,θ,λ) holds. Let A 2 M 1 and A 1 M 1 with A 2 = λ, M 0 = θ, and A 1 = χ; WLOG M 1 = χ. Let M 2 contain A 2 and M 0 be of size λ. By (E) (χ,θ,θ,λ), there is some A 1 such that gtp(a 1 /M 0 ) = gtp(a 1/M 0 ) and A 1 M 2. It will be enough to show that gtp(a 1 A 2 /M 0 ) = gtp(a 1A 2 /M 0 ). By (< κ, λ + χ)-type shortness over θ-sized domains, it is enough to show that, for all a 2 A 2 and corresponding a 1 A 1 and a 1 A 1 of length < κ, we have gtp(a 1 a 2 /M 0 ) = gtp(a 1a 2 /M 0 ). By (M), we have that a 1 M 2 and a 2 M 1, so this follows by Lemma 4.3 above. Now that we have shown the type equality, let f Aut C such that f(a 1 A 2 ) = A 1A 2. Applying f to A 1 M 2, we get that A 1 f(m 2 ) and A 2 = f(a 2 ) f(m 2 ), as desired. (2) Suppose (S) (<κ,θ,<κ). Let A and A be sets of size λ and M 0 N 0 of size θ and χ, respectively, so that gtp(a/m 0 ) = gtp(a /M 0 ) and A N and A N

16 16 WILL BONEY AND RAMI GROSSBERG As above, it is enough to show that gtp(an/m 0 ) = gtp(a N/M 0 ). By type shortness, it is enough to show this for every n N and corresponding a A and a A of lengths less than κ. By (M), we know that a N and a N. By applying (S) (<κ,θ,<κ) to the former, there is N a M 0 containing a such that n N a. As above, Lemma 4.3 gives us the desired conclusion. 5. The main theorem We now state the ideal conditions under which our nonforking works. We reiterate Hypothesis 3.1 in the statement of the theorem for clarity. Theorem 5.1. Let K be an AEC with amalgamation, joint embedding, and no maximal models. If there is some κ > LS(K) such that (1) K is fully < κ-tame and -type short; (2) K doesn t have the weak κ-order property; and (3) satisfies (E) then is an independence relation. Proof: First, by Proposition 4.4, always satisfies (I), (M), (T ), and (C) <κ. Second, (E) is part of the hypothesis. Third, by the other parts of the hypothesis, we can use Proposition 4.1. Let χ, θ, and λ be cardinals. We know that (E) (χ,θ,θ,λ) holds, so (S) (λ,θ,χ) holds. From this, we also know that (S) (<κ,θ,<κ) holds. Thus, (U) (λ,θ,χ) holds. So is an independence relation as in Definition 3.3. In the following sections, we will assume the hypotheses of the above theorem and use as an independence relation. First, we discuss the hypotheses and argue for their naturality. amalgamation, joint embedding, and no maximal models These are a common set of assumptions when working with AECs that appear often in the literature; see [Sh394], [GV06a], and [GVV] for examples. Readers interested in work on AECs without these assumptions are encouraged to see [Sh576] or Shelah s work on good λ-frames in [Sh:h] and [JrSh875]. fully < κ-tame and -type short As discussed in [Bon14b], these assumptions say that Galois types are equivalent to their small approximations. Without this equivalence, there is no reason to think that our nonforking, which is defined in terms of small approximations, would say anything useful about an AEC. On the other hand, we argue that these properties will occur naturally in any setting with a notion of independence or stability theory. The introduction of [GV06a]

17 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 17 observes that this happens in all known cases. Additionally, the following proposition says that the existence of a nonforking-like relation that satisfies stability-like assumptions implies tameness and some stability. Proposition 5.2. If there is a nonforking-like relation that satisfies (U), (M), and κ α ( ) <, then K is (< µ, µ) tame for less than α length types for all regular µ κ α ( ). Proof: Let p q gs <α (M) so their restriction to any smaller submodel is equal and let M i K <µ : i < µ be a resolution of M. By the local character, there are i p and i q such that p does not fork over M ip and q does not fork over M iq. By (M), both of the types don t fork over M ip+iq and, by assumption, p M ip+iq = q M ip+iq. Thus, by (U), we have p = q. The results of [Bon14b, Section 3] allow us to get a similar result for type shortness. The arguments of [MaSh285, Proposition 4.14] show that this can be used to derive stability-like bounds on the number of Galois types. no weak κ order property In first order model theory, the order property and its relatives (the tree order property, etc) are well-studied as the non-structure side of dividing lines. In broader contexts such as ours, much less is known. Still, there are some results, such as Shelah [Sh:e, Chapter III], which shows that a strong order property, akin to getting any desired order of a certain size in an EM model, implies many models. Note that Shelah does not explicitly work inside an AEC, but the proofs and definitions are sufficiently general and syntax free to apply here. Ideally, the weak κ-order property could be shown to imply non-structure for an AEC. While this is not currently known in general, we have two special cases where many models follows by combinatorial arguments and the work of Shelah. First, if we suppose that κ is inaccessible, then we can use Shelah s work to show that there are the maximum number of models in every size above κ. We will show that, given any linear order, there is an EM model with the order property for that order. This implies [Sh:e] s notion of weakly skeleton-like, which then implies many models by [Sh:e, Conclusion III.3.25]. Proposition 5.3. Let κ be inaccessible and suppose K has the weak κ-order property. Then, for all linear orders I, there is EM model M, small N M, p q gs(m), and a i, b i M : i I such that, for all i, j I, i j = gtp(a i b j /M) = p i > j = gtp(a i b j /M) = q Thus, for all χ > κ, K χ has 2 χ nonisomorphic models.

18 18 WILL BONEY AND RAMI GROSSBERG We sketch the proof and refer the reader to [Sh:e] for more details. Proof Outline: Let p q gs(n) and a i, b i : i < κ witness the weak order property. Since K has no maximal models, we may assume that this occurs inside an EM model. In particular, there is some Φ proper for linear orders so N EM(κ, Φ) L that contains a i, b i : i < κ, L(Φ) contains Skolem functions, and κ is indiscernible in EM(κ, Φ) L. Recall that, for X EM(κ, Φ), we have Contents(X) := {I κ : X EM(I, Φ) }. By inaccessibility, we can thin out {Contents(a i b i ) : i < κ} to {Contents(a i b i ) : i J} that is a head-tail system of size κ and are all generated by the same term and have the same quantifier free type in κ. Since κ is regular and Contents(N) is of size < κ, we may further assume the non-root portion of this system is above sup(contents(n)). By the definition of EM models, we can put in any linear order into EM(, Φ) L and get a model in K. Thus, we can take the blocks that generate each a i b i with i J and arrange them in any order desired. In particular, we can arrange them such that they appear in the order given by I. Then, the order indiscernibility implies that the order property holds as desired. We have shown the hypothesis of [Sh:e, Conclusion III.3.25] and the final part of our hypothesis is that theorem s conclusion. We can also make use of these results without large cardinals. To do so, we forget some of the tameness and type shortness our class has to get a slightly weaker relation. Suppose K is < κ tame and type short. Let λ be regular such that λ κ = λ > κ. By the definitions, K is also < λ tame and type short, so take λ to be our fixed cardinal κ. In this case, the ordered sequence constructed in the proof of Lemma 4.3 is actually of size < κ. This situation allows us to repeat the above proof and construct 2 κ non-isomorphic models of size κ. Many other cardinal arithmetic set-ups suffice for many models. (E) We have already mentioned that Existence for κ-coheir is equivalent to the fact every model is κ-galois saturated. The following theorem shows that this follows from categoricity in a κ-closed cardinal. Theorem 5.4. Suppose K is an AEC satisfying the amalgamation property, JEP and has no maximal models. If K is categorical in a cardinal λ satisfying λ = λ <κ, then every member of K χ is κ-galois saturated, where χ = min{λ, sup µ<κ (ℶ (2 µ ) +)}. Proof: First, note that by the assumptions on K and the assumption that λ = λ <κ we can construct a κ-galois saturated member of K λ. Since this class is categorical, all members of K λ are κ-galois saturated. The easy case is when λ < χ: Suppose M K is not κ-saturated and M > λ. Then there is some small M M and p gs(m ) such that p is not realized in M. Then let N M be any substructure of size λ containing M. Then N doesn t realize p, which contradicts its κ-galois saturation.

19 FORKING IN SHORT AND TAME ABSTRACT ELEMENTARY CLASSES 19 For the hard part, suppose M K is not κ-galois saturated and M sup µ<κ (ℶ (2 µ ) +). There is some small M M and p gs(m ) such that p is not realized in M. We define a new class (K +, + ) that depends on K, p and M as follows: L(K + ) := L(K) {c m : m M } by K + = {N : N is an L(K + ) structure st N L(K) K, there exists h : M N L(K) a K-embedding such that h(m) = (c m ) N for all m M and N L(K) omits h(p)}. N 1 + N 2 N 1 L(K) N 2 L(K) and N 1 L(K + ) N 2. This is clearly an AEC with LS(K + ) = M < κ and M, m m M K +. By Shelah s Presentation Theorem K + is a P C µ,2 µ for µ := LS(K + ). By [Sh:c, Theorem VII.5.5] the Hanf number of K + is ℶ (2 µ ) + χ. Thus, K + has arbitrarily large models. In particular, there exists N + K + λ. Then N + L(K) K λ is not κ-galois saturated as it omits its copy of p. Regarding Extension, the strength of this assumption is not entirely clear. The first order version is proved with compactness and the fact that it holds under strongly compact cardinals (see Theorem 8.2) does not work to separate from compactness. However, it does indeed hold in nonelementary classes; see the discussion of quasiminimal classes and λ-saturated models below. Very recently, Vasey [Vasc] is able to show that, if κ = ℶ κ in addition to our other hypotheses and K is λ-galois stable for λ κ instead of no order property and χ-tame for some χ < κ, then Extension holds (and moreover that κ-coheir is an independence relation) for the class of λ + -saturated models. Remark 5.5. While the rest of the results use that K satisfies all of Hypothesis 3.1, the proof of Theorem 5.4 only uses the amalgamation property and also avoid any use of tameness or type shortness. Note that Propositions 5.2 and 5.3 and Theorem 5.4 are not used in the rest of the paper, but are intended to motivate the hypotheses of Theorem 5.1 as natural. At the referee s request, we explain the difficulties in adapting the first-order proof to the current context. The main difficulty is the standard one in trying to transfer results from first-order to AECs: the lack of syntax and compactness. The lack of syntax was helped by the recent isolation of the properties of tameness and type shortness. These properties allow us to treat Galois types as maximal collections of small Galois types, in the same way they are maximal collections of formulas in first-order. However, this does not solve all problems as there are still two key differences: types are only well-behaved over models and types cannot be constructed inductively in an effective manner (i. e., every type is complete over it s domain). Nonetheless, this intuition allows the authors to adapt many arguments from first-order. Regarding compactness, tameness and type

20 20 WILL BONEY AND RAMI GROSSBERG shortness are approximations, but assuming Extension outright is the strongest approximation. Before continuing, we also identify a few contexts which are known to satisfy this hypothesis, especially (1), (2), and (3) of Theorem 5.1. First order theories: Since types are syntactic and over sets, they are < ℵ 0 tame and < ℵ 0 type short and (4) follows by compactness. Additionally, when (3) holds, the theory is stable so coheirs are equivalent to non-forking. Large cardinals: Boney [Bon14b] proves that (1),(2), and Extension hold for any AEC K that are essentially below a strongly compact cardinal κ (this holds, for instance, if LS(K) < κ). Slightly weaker (but still useful) versions of (1) and (2) also hold if κ is measurable or weakly compact. See Section 8 for more. Homogeneous model theory: The sequential homogeneity of the monster model means that Galois types are syntactic, so we have < ℵ 0 tameness and type shortness as above. See, for instance, Grossberg and Lessmann [GrLe02] for more discussion and references. Quasiminimal classes: The quasiminimal closure operation means that Galois types are quantifier free types and amalgamation and other properties are proved in the course of proving categoricity; see [Bal09, p. 190]. Saturated models of a superstable theory: Let T be a superstable first-order theory and KT λ be the class of λ-saturated models of T ordered by elementary substructure. This is a nonelementary class, but still satisfies our hypotheses. Averageable classes: Averageable classes are EC(T, Γ) classes that have a suitable ultraproduct-like relation (that averages the structures), see Boney [Bonb]. Examples of averageable classes are dense ordered group with a cofinal Z-chain and ordered vector fields. The existence of an ultraproduct-like operation mean that (E) (and much more) can be proved similar to the first order version. Amongst these classes, the question of stability in the guise of no order property becomes crucial. Torsion modules over a PID are an example that lie on the stable side of the line. Since the circulation of early drafts, the notion of κ-coheir has been used and extended by various authors, especially Boney, Grossberg, Kolesnikov, and Vasey in [BGKV] and Vasey in [Vasb], [Vasc] and [GVas]. Moreover, [BGKV, Theorem 6.7] has shown that, if κ-coheir is an independence relation, then it is the only independence relation. Theorem 5.6 ( [BGKV]). Under the hypotheses of Theorem 5.1, is the only independence relation on K κ. In particular, if satisfies (I), (M), (E), and (U), then = on K κ.